Characterizations based on record values and order statistics

Consider a sequence of independent and identically distributed random variables {_Xi,i?1}$\{X_i,i?1\}$ with a common absolutely continuous distribution function F  . Let _X1:n?_X2:n???_Xn:n$X_{1:n}?X_{2:n}???X_{n:n}$ be the order statistics of {_X1,_X2,…,_Xn}$\{X_1,X_2,\dots ,X_n\}$ and {_Yl,l?1}$\{Y_l,l?1\}$ be the sequence of record values generated by {_Xi,i?1}$\{X_i,i?1\}$. In this work, the conditional distribution of _Yl$Y_l$ given _Xn:n$X_{n:n}$ is established. Some characterizations of F   based on record values and _Xn:n$X_{n:n}$ are then given.     

Prediction of order statistics and record values based on ordered ranked set sampling

In this paper, we consider two-sample prediction problems. First, based on ordered ranked set sampling (ORSS) introduced by Balakrishnan and Li [Ordered ranked set samples and applications to inference. Ann Inst Statist Math. 2006;58:757–777], we obtain prediction intervals for order statistics from a future sample and compare the results with the one based on the usual-order statistics. Next, we construct prediction intervals for record values from a future sequence based on ORSS and compare the results with the one based on an another independent record sequence developed recently by Ahmadi and Balakrishnan [Prediction of order statistics and record values from two independent sequences. Statistics. 2010;44:417–430].     

Characterizations based on Rényi entropy of order statistics and record values

Two different distributions may have equal Rényi entropy; thus a distribution cannot be identified by its Rényi entropy. In this paper, we explore properties of the Rényi entropy of order statistics. Several characterizations are established based on the Rényi entropy of order statistics and record values. These include characterizations of a distribution on the basis of the differences between Rényi entropies of sequences of order statistics and the parent distribution.     

On characterizations of the generalized Pareto distributions based on progressively censored order statistics

It is well-known, in the literature, that most of the characterization results on exponential distribution are based on the solution of Cauchy functional equation and integrated Cauchy functional equation. In the present paper, we consider the functional equation $$F(x) = F(xy) + F(xQ(y)), \quad x, xQ(y) \in [0, \theta),\; y \in [0,1],$$ where F and Q satisfy certain conditions, to give some new characterization results on the generalized Pareto distributions based on progressively Type-II right censored order statistics. We prove the main results without restricting to distributions that are absolutely continuous with respect to Lebesgue measure.     

On distributions of exceedances associated with order statistics and record values for arbitrary distributions

Received: October 12, 1999; revised version: April 6, 2000     

Some characterizations of geometric tail distributions based on record values

Let R_j be the jth upper record value from an infinite sequence of independent identically distributed positive integer valued random variables. We show that their common distribution must have geometric tail if R_j+k?R_j and R_j are partially independent for some j≥1 and k≥1 or if E(R_j+2?R_j+1| R_j) is a constant. Three versions of partial independence, each of which provides a characterization of the geometric tail are presented.     

Prediction of order statistics and record values from two independent sequences

In this paper, we discuss the problem of predicting future order statistics based on observed record values and similarly, the prediction of future records based on observed order statistics. The coverage probabilities of these intervals are exact and are free of the parent distribution F. Finally, two data sets are used to illustrate the proposed procedures.     

Asymptotic distributions of order statistics and record values arising from the class of beta-generated distributions

In this paper, we show that both the class of beta-generated distributions G_F and its base distribution F belong to the same domain of maximal (or minimal or upper record value or lower record value) attraction. Moreover, it is shown that the weak convergence of any non-extreme order statistic (central or intermediate order statistic), based on a base distribution F, to a non-degenerate limit type implies the weak convergence of G_F to a non-degenerate limit type. The relations between the two limit types are deduced.     

Bayesian prediction of order statistics based on k-record values from exponential distribution

Bayesian prediction of order statistics as well as the mean of a future sample based on observed record values from an exponential distribution are discussed. Several Bayesian prediction intervals and point predictors are derived. Finally, some numerical computations are presented for illustrating all the proposed inferential procedures.     

Moments of order statistics from doubly truncated continuous distributions and characterizations

In this paper, recurrence relations from a general class of doubly truncated continuous distributions which are satisfied by single as well as product moments of order statistics are obtained. Recurrence relations from doubly truncated generalized Weibull, exponential, Raleigh and logistic distributions have been derived as special cases of our result, Some previous results for doubly truncated Weibull, standard exponential, power function and Burr type XII distributions are obtained as special cases. The general recurrence relation of single moments has been used in the case of the left and right truncation to characterize the Weibull, Burr type XII and Pareto distributions.     

Extended exponential distribution based on order statistics

The extended exponential distribution due to Nadarajah and Haghighi (2011 Nadarajah, S., Haghighi, F. (2011). An extension of the exponential distribution. Statistics 45:543–558.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) is an alternative to and always provides better fits than the gamma, Weibull, and the exponentiated exponential distributions whenever the data contain zero values. We establish recurrence relations for the single and product moments of order statistics from the extended exponential distribution. These recurrence relations enable computation of the means, variances, and covariances of all order statistics for all sample sizes in a simple and efficient manner. By using these relations, we tabulate the means, variances, and covariances of order statistics and derive best linear unbiased estimates of the extended exponential distribution. Finally, a data application is provided.     

UMPU tests based on sequential order statistics

Sequential order statistics with conditional proportional hazard rates form a regular exponential family in the model parameters. This finding is used to establish uniformly most powerful unbiased (UMPU) tests for a variety of hypotheses.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Inferences for generalized exponential distribution based on record statistics

In this paper we consider three parameter generalized exponential distribution. Exact expressions for single and product moments of record statistics are derived. These expressions are written in terms of Riemann zeta and polygamma functions. Recurrence relations for single and product moments of record statistics are also obtained. These relations can be used to obtain the higher order moments from those of the lower order. The means, variances and covariances of the record statistics are computed for various values of the shape parameter and for some record statistics. These values are used to compute the coefficients of the best linear unbiased estimators of the location and scale parameters. The variances of these estimators are also presented. The predictors of the future record statistics are also discussed.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Bayesian prediction of order statistics with fixed and random sample sizes based on k-record values from Pareto distribution

In this paper, the two-parameter Pareto distribution is considered and the problem of prediction of order statistics from a future sample and that of its geometric mean are discussed. The Bayesian approach is applied to construct predictors based on observed k-record values for the cases when the future sample size is fixed and when it is random. Several Bayesian prediction intervals are derived. Finally, the results of a simulation study and a numerical example are presented for illustrating all the inferential procedures developed here.     

On the moments of record values

In this paper some general relations for expectations of functions of record values are established. It is seen that these relations may be used to obtain recurrence relations for moments of record values. Bounds on expectations of record values with numerical computations are presented. Applications to the characterizations of the generalizeed exponential distribution are also given.     

Characterization based on generalized entropy of order statistics

ABSTRACT

In the present study, several characterizations of order statistics are obtained on the basis of the generalized entropy. Under some conditions, it is shown that the parent distribution can be uniquely determined by equality of generalized entropy of order statistics.